A Proof on Hypothesis of Dirichlet Divisor Problem

TL;DR

This paper proves a specific asymptotic estimate for the Dirichlet divisor problem, showing the error term behaves as a power of X with a slowly decreasing epsilon.

Contribution

It provides a rigorous proof that the error term in the Dirichlet divisor problem follows a precise asymptotic form involving a logarithmic correction.

Findings

Error term is $X^{1/4+rac{c}{ ext{loglog} X}}$

Confirms conjectured behavior of the divisor problem error term

Advances understanding of divisor function distribution

Abstract

We have proved that $\triangle(X)=X^{1/4+\epsilon(X)}$, $\epsilon(X)=\frac{c}{\log\log X}$.